Think visually

Rise over run: read slope from a graph

Follow horizontal and vertical changes without mixing their order.

“Rise over run” is a visual way to remember slope. Between two points on a straight line, the rise is the signed vertical change and the run is the signed horizontal change. Their quotient tells you how steeply the line changes in coordinate units.

Read a line from left to right

Choose two points where the line passes through known coordinates. Move horizontally from the first x coordinate to the second, then vertically to the second point. If you move 4 units right and 2 units up, the slope is 2/4 = 1/2.

Downward movement gives a negative rise. For 3 units down and 6 units right, the slope is -3/6 = -1/2. The distance traveled along the slanted line is not the run; use only the horizontal change.

Count units, not just grid squares

Some graphs mark the x-axis in steps of 2 and the y-axis in steps of 5. One square right and one square up would then mean a slope of 5/2, not 1. Read the axis labels before counting boxes.

The graph on our calculator uses equal coordinate-unit scales on both axes. It may zoom to the two points, so the origin does not always appear. The exact slope is calculated from the entered coordinates, not estimated from pixels.

Why a larger triangle gives the same slope

Along a straight line, the rise and run scale together. A rise of 2 over a run of 4, and a rise of 5 over a run of 10, both give 1/2. The right triangles are similar. This connection to equivalent ratios helps explain why a line has one constant slope.

A curved graph is different

For a curve, the slope between two points is the slope of the secant line connecting them. It is an average rate of change over that interval, not necessarily the curve's instantaneous slope at either endpoint.

Explore the positive and negative examples in the calculator, then use the sign guide to connect direction with the numerical result. For the algebra behind the picture, read the slope formula.